Harmony earns a $\$42{,}000$ salary in the first year of her career. Each year, she gets a $4\%$ raise. Which expression gives the total amount Harmony has earned in her first $n$ years of her career? Choose 1 answer: Choose 1 answer: (Choice A) A $42{,}000\left(\dfrac{1-0.04^n}{0.96}\right)$ (Choice B) B $42{,}000\left(\dfrac{1-1.04^n}{-0.04}\right)$ (Choice C) C $42{,}000\left(\dfrac{1-1.04^n}{0.96}\right)$ (Choice D) D $42{,}000\left(\dfrac{1-0.96^n}{-0.04}\right)$
Answer: Notice that Harmony's salaries over the years form a geometric sequence. The total amount Harmony earns after $ n$ years is the ${\text{sum}}$ of the first $n$ terms in the sequence. This is called a geometric series. This is the formula for that sum: $ S={a}\left(\dfrac{1-{r}^{ n}}{1-{r}}\right)$ where ${a}$ is the first term and ${r}$ is the common ratio. We can use this formula, along with the given information, to find the expression for the sum, $ S$. Using the given information We are given that Harmony earns a ${\$42{,}000}$ salary in the first year of her career. This is the first term $ a$. We are given that she gets a ${4\% \text{ raise}}$ each year. So we'll use a common ratio of ${1.04}$ for $ r$. We are interested in the first ${n}$ years, so the number of terms is $ {n}$. We want an expression for the total amount she earns. This is the sum $ S$. Writing the sum $ S={42{,}000} \left( \dfrac{1-{1.04}^{{n}}}{1-{1.04}} \right)$ Answer The total amount Harmony has earned in her first $n$ years of her career is: $42{,}000\left(\dfrac{1-1.04^n}{-0.04}\right)$